hw9.Rmd

---
title: "Homework #9"
author: "Ming Chen & Wenqiang Feng"
date: "2/2/2017"
output: html_document
---

## You can also get the [PDF format](./pdf/HW9.pdf) of Wenqiang's Homework.

## Data

```{R}
## response variable: PULSE
## predictor variables: RUN, SMOKE, HEIGHT, WEIGHT, PHYS1, PHYS2
mlrdata = read.table("./data/ex13-69.TXT", header = T, row.names = 1)
mlrdata$RUN = as.factor(mlrdata$RUN)
mlrdata$RUN = relevel(mlrdata$RUN, ref = "0")
mlrdata$SMOKE = as.factor(mlrdata$SMOKE)
mlrdata$SMOKE = relevel(mlrdata$SMOKE, ref="0")
mlrdata$PHYS1 = as.factor(mlrdata$PHYS1)
mlrdata$PHYS1 = relevel(mlrdata$PHYS1, ref="0")
mlrdata$PHYS2 = as.factor(mlrdata$PHYS2)
mlrdata$PHYS2 = relevel(mlrdata$PHYS2, ref="0")
```

## problem a

```{R}
mlrfit = lm(PULSE~RUN+SMOKE+HEIGHT+WEIGHT+PHYS1+PHYS2, data=mlrdata)
summary(mlrfit)
beta0 = coef(mlrfit)['(Intercept)']
betaRun1 = coef(mlrfit)['RUN1']
betaSmoke1 = coef(mlrfit)['SMOKE1']
betaHeight = coef(mlrfit)['HEIGHT']
betaWeight = coef(mlrfit)['WEIGHT']
betaPhys1 = coef(mlrfit)['PHYS11']
betaPhys2 = coef(mlrfit)['PHYS21']
```

Full Model: PULSE = `r beta0` + `r betaRun1`\*(RUN=1)  `r betaSmoke1`\*(SMOKE=1)  `r betaHeight`\*(HEIGHT) + `r betaWeight`\*(WEIGHT) + `r betaPhys1`\*(PHYS1) + `r betaPhys2`\*(PHYS2)

## problem b

```{R fig.width=5, fig.height=5}
rstudentVal = rstudent(mlrfit)
cooksDisVal = cooks.distance(mlrfit)
plot(rstudentVal~cooksDisVal, xlab="Cook's Distance", 
     ylab="Studentized Residuals", ylim=c(-3,3))
abline(h=c(-2,2), lty="dashed", lwd=2, col="blue")
potentialInfObs = cooksDisVal[rstudentVal>2 | rstudentVal<(-2)]
potentialInfObs
```

No influential observations exit. There is one observation that has large studentized residual. However, Cook's D for this observation is `r potentialInfObs`, which indicates small influence on the overall fit of the model.

#### problem c

```{R}
stepwiseFit = step(mlrfit, direction = "both", test="F")
```

Stepwise regression selection indicates that *RUN*, *SMOKE*, *PHYS1* and *PHYS2* are important variables to the model.

#### problem d

Choose the four important variables to re-fit the model.

```{R}
finalMod = lm(PULSE~RUN+SMOKE+PHYS1+PHYS2, data=mlrdata)
summary(finalMod)

beta0 = coef(finalMod)['(Intercept)']
betaRun1 = coef(finalMod)['RUN1']
betaSmoke1 = coef(finalMod)['SMOKE1']
betaPhys1 = coef(finalMod)['PHYS11']
betaPhys2 = coef(finalMod)['PHYS21']
```

The result shows that only variable *RUN*, *SMOKE*, *PHYS1* have significant effect on *PULSE*. 

* Final Model: PULSE = `r beta0` + `r betaRun1`\*(RUN=1)  `r betaSmoke1`\*(SMOKE=1) + `r betaPhys1`\*(PHYS1)  

    + Given that other variables are consistent, the pulse of people who ran was 11.132 higher than people who didn't run.
    + Given that other variables are consistent, the pulse of people who smoke was 6.963 lower than people who don't smoke.
    + Given that other variables are consistent, the pulse of people who did a lot of physical exercise is 13.325 higher than people who didn't.